Abstract
Instrumental variable (IV) analysis addresses bias owing to unmeasured confounding when comparing two nonrandomized treatment groups. To date, studies in the statistical and biomedical literature have focused on the local average treatment effect (LATE), the average treatment effect for compliers. In this article, we study the weighted local average treatment effect (WLATE), which represents the weighted average treatment effect for compliers. In the WLATE, the population of interest is determined by either the instrumental propensity score or compliance score, or both. The LATE is a special case of the proposed WLATE, where the target population is the entire population of compliers. Here, we discuss the interpretation of a few special cases of the WLATE, identification results, inference methods, and optimal weights. We demonstrate the proposed methods with two published examples in which considerations of local causal estimands that deviate from the LATE are beneficial.
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Acknowledgements
This research was supported in part by the National Cancer Institute for the Mays Cancer Center (P30CA054174) at the UT Health Science Center at San Antonio.
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Appendices
Appendix A: Proof for Eq. (11)
Let \(f_z(x)=\text{ pr }(X=x\mid Z=z)\), which is equal to \(f(x)\text{ pr }(Z=z\mid X=x)/\text{pr }(Z=z)\). Based on Bayes’ theorem, for \(z=\{0,1\}\), \(f_z^c(x) = \text{ pr }(X=x\mid Z=z, U=c)\) can be written as
Equation (A1) holds because U is independent of Z conditional on X by Assumption 1. Using Bayes’ theorem, \(f^c(x) = \text{ pr }(U=c\mid X=x)f(x)/\text{pr }(U=c)\). Therefore, Eq. (A2) becomes
which implies Eq. (11).
Appendix B: Proof for Theorem 2
We seek to determine h(x) that minimizes
Let \(k(X) = \sigma ^{2}_{h,1}(X)/e(X) + \sigma ^{2}_{h,0}(X)/\{1-e(X)\}\) and \(g(X) = \delta (X)h(X)\). Then we attempt to find g(x) that minimizes
We can normalize g(x) to satisfiy \(\int g(x)f(x)dx=1\). Then, our problem becomes the minimization of
The solution should satisfy \(0 = 2\,g(x) \{k(x)/\delta (x)^2\} f(x) - \lambda f(x)\); thus, the solution g(x) is proportional to \(\delta (x)^2/k(x)\). Therefore, the solution of h(x) is \(\delta (x)/k(x)\).
Appendix C: Proof for Theorem 3
Under the conditions of Theorems 1 and 3, based on the results of Section 4.2. of Hirano et al. (2003), we have \(\sqrt{n}({\hat{\tau _h}} - \tau _h) = (1/\sqrt{n})\sum _{i=1}^n t(Y_i,D_i,Z_i,X_i)+o_p(1)\) and \(\sqrt{n}({\hat{\delta _h}} - \delta _h) = (1/\sqrt{n})\sum _{i=1}^n \pi (Y_i,D_i,Z_i,X_i)+o_p(1)\), where
A first-order Taylor expansion of \({\hat{\tau _h}}/{\hat{\delta _h}}\) around the point \((\tau _h, \delta _h)\) yields
Applying Eqs. (C1) and (C2) to Eq. (C3) gives
Applying the Lindeberg–Levy central limit theorem to Eq. (C4) gives the asymptotic normal result in Theorem 3.
Appendix D: Proof for Theorem 5
It suffices to show that \({\hat{\tau _h}}(1)\) and \({\hat{\tau _h}}(0)\) are consistent estimators for \(\tau _h(1)\) and \(\tau _h(0)\). The following equalities hold because \(Y = DY(1)+(1-D)Y(0)\), \(D=ZD(1)+(1-Z)D(0)\), and Z is independent of all potential outcome and treatment variables conditional on X.
Hence, we can write \(\tau _h(1)\) as
Therefore, we obtain the following consistent estimator for \(\tau _h(1)\):
We can show that \(Z_iD_i(1)Y_i(1) = Z_iD_iY_i\) and \((1-Z_i)D_i(0)Y_i(1) = (1-Z_i)D_iY_i\). This gives the estimator \({\hat{\tau _h}}(1)\) in Eq. (18).
We can write \(\tau _h(0)\) as
Therefore, we obtain the following consistent estimator for \(\tau _h(0)\):
It can be shown that \((1-Z_i)(1-D_i(0))Y_i(0) = (1-Z_i)(1-D_i)Y_i\) and \(Z_i(1-D_i(1))Y_i(0) = Z_i(1-D_i)Y_i\). This gives the estimator \({\hat{\tau _h}}(0)\) in Eq. (19).
Appendix E: Figures
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Choi, B.Y. Instrumental variable estimation of weighted local average treatment effects. Stat Papers 65, 737–770 (2024). https://doi.org/10.1007/s00362-023-01415-2
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DOI: https://doi.org/10.1007/s00362-023-01415-2
Keywords
- Compliance scores
- Instrumental variables
- Local average treatment effects
- Weighted local average treatment effects